Interferometric polarization control

ABSTRACT

A signal conditioning module provides a polarimeter capability in a photometric system. The module may include multiple variable delay polarization modulators. Each modulator may include an input port, and a first arm formed to include a first reflector and first rooftop mirror arranged in opposed relationship. The first reflector may direct an input radiation signal to the first rooftop mirror. Each modulator also may include an output port and a second arm formed to include a second reflector and second rooftop mirror arranged in opposed relationship. The second reflector can guide a signal from the second rooftop mirror towards the output port to provide an output radiation signal. A beamsplitting grid may be placed between the first reflector and the first rooftop mirror, and also between the second reflector and the second rooftop mirror. A translation apparatus can provide adjustment relative to optical path length vis-à-vis the first arm, the second arm and the grid.

RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Application Ser.No. 60/692,713, filed on Jun. 20, 2005, wider 35 U.S.C. §119(e).

This application is a divisional application and claims the benefit ofU.S. Non-Provisional application Ser. No. 11/425,352, filed Jun. 20,2006.

ORIGIN OF THE INVENTION

The invention described herein was made by one or more employees of theUnited States Government and may be manufactured and used by or for theGovernment of the United States of America for governmental purposeswithout the payment of any royalties thereon or therefor.

INCORPORATION BY REFERENCE

This application incorporates by reference U.S. application Ser. No.11/425,352, filed Jun. 20, 2006.

FIELD OF THE DISCLOSURE

This disclosure relates generally to measurement apparatus, modeling andtechniques for polarization characterization or measurement of signals,in particular to a quasioptical phase modulator for polarization controland modulation, and more particularly, to techniques and apparatus forvariable delay polarization modulation.

BACKGROUND

Astronomical polarimetry is an area currently undergoing much study anddevelopment, at least in part responsive to high-sensitivity searchesfor “B-modes” of the cosmic microwave background radiation polarization.

The cosmic microwave background is polarized at the level of a fewmicrokelvins. There are two types of polarization, respectively known asE-modes and B-modes. The relationship between these modes may beanalogous to electrostatics, in which the electric field (E-field) has avanishing curl and the magnetic field (B-field) has a vanishingdivergence. The E-modes arise naturally from Thomson scattering ininhomogeneous plasma. The B-modes, which are thought to have anamplitude/magnitude of at most a 0.1 μK, are not produced from plasmaphysics alone.

Detecting the B-modes is extremely difficult, particularly given thatthe degree of foreground contamination is unknown, and the weakgravitational lensing signal mixes the relatively strong E-mode signalwith the B-mode signal.

B-modes are signals resulting from cosmic inflation and are determinedby the density of primordial gravitational waves. B-modes thus providesignatures for gravitational waves associated with the inflationaryepoch and are expected to provide a direct measurement of the energyscale of inflation. Amplitudes for B-modes are theorized to be on theorder of 10⁻⁷ to 10⁻⁹ of that of the cosmic background radiation, andthus measurement of the B-modes requires a robust modulation strategyand effective control over systematic artifacts.

Emission from magnetically-aligned dust in our Galaxy contributes tointerference that will have to be understood in order to clearlydistinguish and extract the B-mode from the total signal. However, thispolarized emission also provides a tool for analyzing the role ofmagnetic fields in star formation. The advent of multiple wavelengthsubmillimeter and far-infrared photometers, such as SCUBA2 (a newgeneration submillimeter imager for the James Clerk Maxwell Telescope)and HAWC (a far-infrared camera for the Stratospheric Observatory ForInfrared Astronomy (SOFIA)), provides opportunity to expand such study.Polarization modules have been developed to facilitate leveraging ofthese new photometric tools for such applications by allowing them tofunction as polarimeters.

Partial polarization results from statistical correlation between theelectric field components in the plane perpendicular to the propagationdirection. Such correlation is represented via complex quantities, and,as a result, in measurements of polarized light, it is convenient toemploy linear combinations of these correlations, such as Stokesparameters, e.g., I, Q, U and V.

The polarization state of radiation through an optical system may bemodeled by determining the transformations that describe the mapping ofthe input to the output polarization states. In modeling the types ofoptical elements associated with the polarization modules described inthis disclosure, Stokes I is decoupled from the other Stokes parameters.For this class of elements, the polarization P, as described withreference to Eq. (1) below,P ² =Q ² +U ² +V ²,  (1)is constant. Eq. (1) may be interpreted to describe the points on thesurface of a sphere in three-dimensional space having Q, U and V ascoordinate axes. This sphere is known as the Poincaré sphere, and theaction of any given ideal polarization modulator may be described by arotation and/or an inversion in this space. Such operations correspondto introduction of a phase delay between orthogonal polarizations, andthat is the physical mechanism operative in a polarization modulator.The two degrees of freedom of any given transformation are the magnitudeof the introduced phase delay and a parameter describing the basis usedto define the phase delay. These two parameters directly define theorientation and the magnitude of the rotation on the Poincaré sphere:the rotation axis is defined by the sphere diameter connecting the twopolarization states between which the phase is introduced, and themagnitude of the rotation is equal to that of the introduced phase.

In order to measure the polarized part of a partially-polarized signal,it is useful to separate the polarized portion of the signal from theunpolarized portion. This is especially useful when the fractionalpolarization is small. One way to do this is to methodically change, ormodulate, the polarized portion of the signal (by changing one of theparameters of the polarization modulator) while leaving the unpolarizedportion unaffected. Periodic transformations in Poincaré space canaccomplish this encoding of the polarized portion of the signal forsubsequent demodulation and detection. A convenient way of formulatingthe problem is to envision a detector that is sensitive to Stokes Q whenprojected onto the sky in the absence of modulation. The polarizationmodulator is then systematically changing the polarization state towhich the detector is sensitive. By measuring the output signal, thepolarization state of the signal or light may be completelycharacterized.

One conventional way to implement such a polarization modulator is byuse of a dielectric birefringent plate. A birefringent plate comprises apiece of birefringent material cut so as to delay one linearpolarization component relative to the other by the desired amount(generally either one-half or on quarter of the wavelength of interest).In this case, the phase difference is fixed, and the modulation isaccomplished by physically rotating the birefringent plate (and hencethe basis of the introduced phase).

However, a birefringent plate may be built to measure either circular orlinear polarization, but cannot measure both. Additionally, polarizationmodulators built using this approach are not readily retuned for use atmultiple wavelengths. Further, the requirement to be able to rotate thebirefringent plate engenders need for a complex ensemble of shafts,bearings and gears.

For the reasons stated above, and for other reasons discussed below,which will become apparent to those skilled in the art upon reading andunderstanding the present disclosure, there are needs in the art toprovide improved phase modulators in support of increasingly stringentand exacting performance and measurement standards in settings such asastronomical observation.

SUMMARY

The above-mentioned shortcomings, disadvantages and problems areaddressed herein, which will be understood by reading and studying thefollowing disclosure.

In one aspect, the disclosure encompasses a signal conditioning moduleconfigured for insertion into a photometric system to realize apolarimeter. The module may include a cascaded series of variable delaypolarization modulators. Each modulator of the series includes mayinclude an input port, and a first arm formed to include a firstreflector and first rooftop mirror arranged in opposed relationship. Thefirst reflector can direct an input radiation signal from the input porttowards the first rooftop mirror. Each modulator may also include anoutput port and a second arm formed to include a second reflector andsecond rooftop mirror arranged in opposed relationship. The secondreflector can guide a signal from the second rooftop mirror towards theoutput port to provide an output radiation signal. A beamsplitting gridmay be placed between the first reflector and the first rooftop mirror,and may also be placed between the second reflector and the secondrooftop mirror. A translation apparatus can provide adjustment relativeoptical path length vis-a-vis the first arm, the second arm and thegrid.

In another aspect, a millimeter wave receiver may include an inputantenna, and a polarization modulation chain coupled to the inputantenna. The modulation chain can include multiple cascadedvariable-delay polarization modulators, each having an input realizing afixed basis of phase introduction, and a translation apparatus providingvariable magnitude of phase delay between two orthogonal linearpolarizations of an input signal. The modulation chain may furtherinclude a beamsplitter, a first arm and a second arm optically coupledto one another via the beamsplitter. Each of the first and second armsmay include opposed reflective elements disposed on opposite sides ofthe beamsplitter. The opposed reflective elements in each arm may haveat least one mirror coupled to the translation apparatus such that afirst distance separating the beamsplitter from at least one of theopposed reflective elements is variable.

In a further aspect, the present disclosure contemplates a radiotelescope, which may comprise an antenna and a receiver coupled to theantenna. The receiver can provide a polarizing interferometer functionby including a polarization modulation chain coupled to the antenna andcomprising multiple cascaded variable-delay polarization modulators,each of which may include a translation apparatus providing variablemagnitude of phase delay between two orthogonal linear polarizations ofan input signal, a beamsplitter and a first arm and a second armoptically coupled to one another via the beamsplitter. Each of the firstand second arms may be formed of opposed reflective elements disposed onopposite sides of the beamsplitter, each of which may include at leastone mirror coupled to the translation apparatus such that a firstdistance separating the beamsplitter from at least one of the opposedreflective elements is variable.

BRIEF DESCRIPTION OF THE DRAWING

FIG. 1 is a simplified block diagram of an overview of a systemconfigured to improve quasi-optical signal detection operationsaccording to an embodiment of the invention.

FIG. 2 is a simplified representation of a variable-delay polarizationmodulator element useful in the context of the system of FIG. 1according to an embodiment.

FIG. 3 is a simplified representation of a variable-delay polarizationmodulator element useful in the context of the system of FIG. 1according to an embodiment.

FIG. 4 is a simplified block diagram illustrating a variable-delaypolarization modulator useful in the context of the system of FIG. 1according to an embodiment.

FIGS. 5 through 9 are graphical depictions of measured and simulatedperformance of a millimeter-wave or quasi-optical signal detectionapparatus using the concepts developed with respect to thevariable-delay polarization modulator element(s) according to anembodiment.

DETAILED DESCRIPTION

In the following detailed description, reference is made to theaccompanying drawings that form a part hereof, and in which are shown,by way of illustration, specific embodiments which may be practiced.These embodiments are described in sufficient detail to enable thoseskilled in the art to practice the embodiments, and it is to beunderstood that other embodiments may be utilized, and that logical,mechanical, electrical and other changes may be made, without departingfrom the scope of the embodiments.

Ranges of parameter values described herein are understood to includeall subranges falling therewithin. The following detailed descriptionis, therefore, not to be taken in a limiting sense.

The detailed description is divided into five sections. In the firstsection, a system level overview is provided. In the second section, aphysical example of a variable-delay polarization modulator element ispresented. In the third section, tools for modeling variable-delaypolarization modulator elements are developed. In the fourth section,hardware and an operating environment in conjunction with whichembodiments may be practiced are described. In the fifth section, aconclusion of the detailed description is provided. A technical effectof the systems and processes disclosed herein includes at least one offacilitating capability for measurement of polarization components ofmillimeter-wave or quasi-optical signals.

§I. System Overview

FIG. 1 is a simplified diagram of an overview of a modified system 100configured to improve quasi-optical signal detection operations. Thesystem 100 may receive a signal 102 from a source 103 via one or moreantennae 105 (only one antenna is shown in FIG. 1), which may beparabolic, or may be any other type of antenna suitable for thewavelength of interest. A receiver 110 can be coupled to the antenna 105and can provide received signals 102 via a signal port 115 to a firstvariable-delay polarization modulator element VPM 120, which may becoupled via signal path 122 to a second variable-delay polarizationmodulator element VPM 124, which, in turn, can pass the modulated signalon to other receiver elements (not shown in FIG. 1) via an output port125.

It will be appreciated that while two variable-delay polarizationmodulator elements VPM 120, 124 are shown coupled in cascade in FIG. 1,additional variable-delay polarization modulator elements. In oneembodiment, the system 100 may be included in the receiver 110. As well,one or more of the variable-delay polarization modulator elements mayintegrate into the system 100 in other ways, such as forming a portionof the antenna 105, or as coupling the antenna 105 to the receiver 110.Also, it will be appreciated that the receiver 110 may be employed incapacities not associated with a signal reception system per se. Forexample, the cascaded variable-delay polarization modulator elements VPM120, 124 may be employed as a portion of a calibration apparatus or forother purposes.

In one embodiment, the system 100 may be a radio reception systemconfigured to detect modulation present on a radio signal 102. Suchmodulation may result from a variety of sources, including man-madesources or naturally-occurring objects and associated phenomena. Inother words, modulation present on the signals 102 may represent any ofa variety of phenomena, including communications-related activities,remote sensing applications, naturally-occurring objects and associatedphenomena and other scenarios wherein detection of such modulation isdesirable. In one embodiment, the system 100 may be at least a portionof an astronomical instrument, such as a radio telescope, that may beemployed for purposes of cosmology or other study.

The variable-delay polarization modulator elements 120 may stand incontrast to the conventional birefringent modulators in that the basisof phase introduction can be held fixed, but the magnitude of the delaymay be variable. As a result, a variable-delay polarization modulatorelement 120 may provide an adjustable delay between two orthogonallinear polarizations of an electromagnetic wave comprising the signal102. There are many known elements in the art capable of providing suchvariable delay. For example, a Martin-Puplett interferometer that doesnot include an input polarizer has been described in the past.

In this invention, the polarization modulation can be explicitlyseparated from the polarized detection (obtained via an analyzer) of thesignal 102. One advantage realized by this treatment may be that thebasis of the variable-delay polarization modulator can be rotated at anarbitrary angle with respect to the analyzer.

§II. Variable-Delay Polarization Modulator Elements

FIG. 2 shows a simplified schematic diagram of a variable-delaypolarization modulator 200 of a Martin-Puplett interferometer, i.e., ann=1 case, for an angle of π/4. The variable-delay polarization modulator200 may include an input port 202, and output port 204, a first rooftopmirror 205 comprising a first planar reflective surface 207 and a secondplanar reflective surface 209 at right angles to one another, a secondrooftop mirror 210 comprising a first planar reflective surface 212 anda second planar reflective surface 214 also at right angles to oneanother and a wire grid 215 acting as a beamsplitting reflector and as apolarizing filter interposed between the first 205 and second 210rooftop mirrors. When d₁=d₂, the modulator 200 may behave as a mirror;when there is a path difference, it can change the polarization state ofthe incoming radiation 220.

An incoming light beam 220 can enter through the input port 202 and thenbe incident on the grid 215. A portion 222 of the beam may betransmitted through the grid 215 and may strike the first planarreflective surface 212 of the second rooftop mirror 210, giving rise toa first reflected beam 226 that may strike the second planar reflectivesurface 214 of the second rooftop mirror 210 and in turn gives rise to asecond reflected beam 228 that may travel parallel to the transmittedbeam portion 222, but in the opposite direction. The second reflectedbeam 228 may strike the grid 215, giving rise to a third reflectedportion 230 that may travel parallel to the first reflected beam 226,but in the opposite direction, to the output port 204.

Another portion of the incoming light beam 220 may be reflected by thegrid 215, giving rise to a first reflected beam 236 that can strike thefirst planar surface 207 of the first rooftop mirror 205, thus resultingin a second reflected beam 238 that can strike the second planar surfaceof the first rooftop mirror 205, and in turn giving rise to a thirdreflected beam 240 traveling in the opposite direction. According to theexample of an embodiment shown in FIG. 2, radiation may enter from theleft (arrow 220), and may be split by a beam-splitting grid 215 intobeams 222 and 236. These two components of polarization may then bedirected to a respective one of rooftop mirrors 205 and 210, which canrotate the polarization by ninety degrees with respect to the wiresforming the beam-splitting grid 215.

Beams 230 and 240 can recombine at the beam-splitting grid 215 and mayexit towards the top of the variable-delay polarization modulator 200.In the following analysis, the angle of the variable-delay polarizationmodulator 200 corresponds to the angle of the beam-splitting grid 215.However, one skilled in the art will recognize that other possibilitiesexist that may fall within the scope of this invention.

The variable-delay polarization modulators 200 can be configured asfollows: the variable-delay polarization modulator closest to thepolarization-sensitive detector, i.e., variable-delay polarizationmodulator₁, may have its beam-splitting grid 215 oriented at forty-fivedegrees with respect to the axis of the detector (Q axis), and thesecond variable-delay polarization modulator, i.e., variable-delaypolarization modulator₂, may have its grid 215 oriented at an angle oftwenty-two and a half degrees (i.e., π/8) with respect to the detectoraxes. Full modulation of all linear and circular polarizations statescan be achieved with this configuration. Employing this architecture ina polarimeter that measures linear polarization may be modeled asdescribed below. Denoting a first variable-delay polarization modulator,i.e., variable-delay polarization modulator₁, which may be an inputvariable-delay polarization modulator and thus near the signal source,as having zero phase delay, and then switching a second variable-delaypolarization modulator, i.e., variable-delay polarization modulator₂,between delays of 0 and π, then the detector axes, projected onto theplane of the sky, can switch between Q and −Q. With variable-delaypolarization modulator₁ set to a phase delay of π, switchingvariable-delay polarization modulators between 0 and π can project thedetector axes to ±U. The dual variable-delay polarization modulators canprovide two degrees of freedom, namely, the phase delay of each. Theangles chosen for the two basis sets may be those for which the twodegrees of freedom correspond to orthogonal coordinates on the Poincarésphere, thus allowing all polarization states to be accessible to thedetector.

Qualities providing utility for this architecture as a candidatetechnology for polarimeters in far-infrared through millimeter regionsof the spectrum for astronomical observation applications may include:(i) this approach may allow implementations that are able to the firstreflected beam 236 and parallel to it. The third reflected beam 240combines with the third reflected beam portion 230 and the combinedbeams 230 and 240 exit towards the top of the variable-delaypolarization modulator 200.

The polarizations of the electric E and magnetic H components of thevarious beam portions can be as noted in FIG. 2, and the rooftop mirrors205 and 210 can act to rotate the polarization by ninety degrees withrespect to the wires forming the beam-splitting grid 215. The pathlength of either the arm of the modulator 200 including the firstrooftop mirror 205 or the second rooftop mirror 210 may be modified, forexample by translating the first rooftop mirror 205 parallel to the pathnoted by the arrow 250, resulting in modulation of relative phase of thethird reflected beam 240 relative to the third reflected beam portion230 by changing the path length d₁, or by translating the second rooftopmirror 210 parallel to the arrow 260, again resulting in controlledmodulation of the relative phases of the third reflected beam portion230 relative to the third reflected beam 240 by changing the path lengthd₂. In either case, the third reflected beam portion 230 and the thirdreflected beam 240 may combine to provide either constructive ordestructive interference varying with the relative path lengthmodulation of d₁-d₂. When d₁=d₂, the modulator 200 behaves as a mirror;when there is a path difference, the polarization state of the incomingradiation 220 is modified as described infra.

FIG. 3 is a simplified representation of a variable-delay polarizationmodulator element 300 useful in the context of the embodiment of FIG. 1.The variable-delay polarization modulator element 300 may include aninput port 302, an output port 304, a mirror 305 and a polarizing wiregrid 315 that also acts as a partially reflective surface or beamsplitter. An input signal 320 (solid line and arrow) may enter via theinput port 302 and strike the grid 315, giving rise to a partiallytransmitted portion 322 (dashed line and arrow) and a partiallyreflected beam portion 330 (dot-dash line and arrow). The partiallytransmitted beam portion 322 may be incident on and reflected by themirror 305, resulting in a reflected beam 340, at least a portion ofwhich may traverse back through the grid 315 parallel to the reflectedbeam portion 330 and which may combine with the reflected beam portion330 at the output port 304. Motion of the mirror 305 relative to thegrid 315 or vice-versa may result in a variable path length differencefor the beam portions 330 and 340, again giving rise to modulatedconstructive or destructive interference, as described above.

In the following analysis, the angle of the variable-delay polarizationmodulator 200 may correspond to the angle of the beam-splitting grid215. In contexts such as the receiver 110 of FIG. 1, where multiplevariable-delay polarization modulators 200 are coupled in cascade, thevariable-delay polarization modulators 200 may be configured as follows:the variable-delay polarization modulator 200 closest to thepolarization-sensitive detector, i.e., variable-delay polarizationmodulator₁, may have its beam-splitting grid 215 oriented at forty-fivedegrees with respect to the axis of the detector (Q axis), and thesecond variable-delay polarization modulator 200, i.e., variable-delaypolarization modulator₂, may have its grid 215 oriented at an angle oftwenty-two and a half degrees (i.e., π/8) with respect to the detectoraxes. Full modulation of all linear and circular polarizations statescan be achieved with this configuration. Employing this architecture ina polarimeter that measures linear polarization may be modeled asdescribed below. Denoting a first variable-delay polarization modulator,i.e., variable-delay polarization modulator, which is an inputvariable-delay polarization modulator and thus near the signal source,as having zero phase delay, and then switching a second variable-delaypolarization modulator, i.e., variable-delay polarization modulator,between delays of 0 and π, then the detector axes, projected onto theplane of the sky, will switch between Q and −Q. With variable-delaypolarization modulator₁ set to a phase delay of π, switchingvariable-delay polarization modulator₂ between 0 and π projects thedetector axes to ±U. The dual variable-delay polarization modulators mayprovide two degrees of freedom, namely, the phase delay of each. Theangles chosen for the two basis sets may be those for which the twodegrees of freedom correspond to orthogonal coordinates on the Poincarésphere, thus allowing all polarization states to be accessible to thedetector.

Qualities providing utility for this architecture as a candidatetechnology for polarimeters in far-infrared through millimeter regionsof the spectrum for astronomical observation applications include: (i)this approach allows implementations that are able to cover the fullPoincaré sphere, in contrast to birefringent plates, which can assesseither linear or circular polarization, but not both; (ii) thevariability of the path difference between orthogonal polarizationstates may facilitate retuning for use at multiple wavelengths; (iii)frequency-dependent antireflective coatings can be avoided, in contrastto approaches relying on transmission through thick dielectric plates;(iv) the complexities of shafts, bearings and gears to rotate thebirefringent plate in modulators based on such plates can be avoided.Each of these qualities may present a benefit in the context ofmeasuring polarized flux of astronomical and cosmological sources fromspace-borne observation platforms.

The variable-delay polarization modulator in a Martin-Puplettinterferometer is a specific example of more general cases. TheMartin-Puplett interferometer is configured with the relative angle ofthe variable-delay polarization modulator at forty-five degrees withrespect to the analyzer. The analytical expressions for the polarizationcomponents in this case may follow from those for the case of the singlevariable-delay polarization modulator placed at an arbitrary angle,although the physical implementation is different.

The development and disclosure of aspects of expressions for thetransfer function for multiple variable-delay polarization modulatorelements in cascade at arbitrary relative orientations may proceed froman example of two such elements, as described in more detail below.Within that context, initially developing an expression for the transferfunction for a single variable-delay polarization modulator then latermay facilitate modeling of multiple cascaded variable-delay polarizationmodulators having arbitrary relative orientations. The discussion thatfollows assumes that the modulation passband is sufficiently narrow thatthe phase delays introduced at the center wavelength approximately applyover the whole bandwidth.

§III. Modeling Variable-Delay Polarization Modulator Elements

A Mueller matrix representation (e.g., Eqs. (2) through (10) and Table1, infra) of the variable-delay polarization modulator 200, according toan embodiment, is now developed in §III(a), based on consideration ofthe interior portion of a Martin-Puplett interferometer. Via thisrepresentation, which is given strictly by way of example and istherefore not limiting, the frequency-dependent performance of avariable-delay polarization modulator can be modeled in §III(b) (e.g.,Eqs. (11) through (16), infra). An alternative embodiment ofarchitecture for the variable-delay polarization modulator is describedin §III(c), and systematics are considered in §III(d). Experimentalresults from laboratory tests of a single variable-delay polarizationmodulator are then presented in §III(e) (and Eq. 17), and thoseempirical results are compared in §III(f) to simulations based on thetransfer function models developed and disclosed herein (FIGS. 5 through9). In §III(g), polarization matrix methods are described (e.g., Eqs. 18through 24 and Table II, infra). A summary of these aspects is providedin §III(h).

A Martin-Puplett interferometer includes a variable-delay polarizationmodulator with an analyzer on the output end nominally oriented at anangle of forty-five degrees with respect to the beam-splitting grid 215.For spectrometer applications, one of the ports on the input side isshorted by a grid 215 oriented either parallel or perpendicular to theanalyzer (a grid 215 may have functionality other than beamsplitting).The model developed below of the interior of the Martin-Puplettinterferometer in terms of Jones and Mueller matrices allows a generalvariable-delay polarization modulator to be described at an arbitraryangle with respect to the optical system within which it is operative.The convenience of use of standard polarization matrices facilitatesmodeling of multiple variable-delay polarization modulators cascadedserially.

Initially, a model is derived for the simple case of a variable-delaypolarization modulator at an angle of forty-five degrees. This simplemodel is then generalized via similarity transformation.

§III(a). Mueller Matrix Representation of a Single Stage

For the simple case, the Jones matrix J _(VPM)(π/4) representing thisconfiguration can be expressed as a sum of Jones matrices for theradiation in each of the arms of the variable-delay polarizationmodulator 200, that is:J _(VPM)(π/4)= J _(VPM) ⁽¹⁾(π/4)+ JVPM ⁽²⁾(π/4).  (2)

In turn, each of the terms can be decomposed into a product of the Jonesmatrices of the elements in each optical path, for example, using theJones matrices given below in Table 1. A result of decomposing the termsof Eq. (2) as described is shown below in Eqs. (3) and (4):

$\begin{matrix}\begin{matrix}{{{\overset{\_}{J}}_{VPM}^{(1)}\left( {\pi/4} \right)} = {{{\overset{\_}{J}}_{WT}\left( {\pi/4} \right)}{{\overset{\_}{J}}_{z}\left( d_{1} \right)}{{\overset{\_}{J}}_{RT}(0)}{{\overset{\_}{J}}_{z}\left( d_{1} \right)}{{\overset{\_}{J}}_{WR}\left( {\pi/4} \right)}}} \\{{= {\begin{pmatrix}1 & 1 \\{- 1} & {- 1}\end{pmatrix}\frac{\exp\left( {{\mathbb{i}4\pi}\;{d_{1}/\lambda}} \right)}{2}}},{and}}\end{matrix} & (3) \\\begin{matrix}{{{\overset{\_}{J}}_{VPM}^{(2)}\left( {\pi/4} \right)} = {{{\overset{\_}{J}}_{WR}\left( {\pi/4} \right)}{{\overset{\_}{J}}_{z}\left( d_{2} \right)}{{\overset{\_}{J}}_{RT}(0)}{{\overset{\_}{J}}_{z}\left( d_{2} \right)}{{\overset{\_}{J}}_{WT}\left( {\pi/4} \right)}}} \\{= {\begin{pmatrix}1 & {- 1} \\1 & {- 1}\end{pmatrix}{\frac{\exp\left( {{\mathbb{i}4\pi}\;{d_{1}/\lambda}} \right)}{2}.}}}\end{matrix} & (4)\end{matrix}$

TABLE 1 Summary of physical transformation models for optical elements,including Jones matrices. For linear distance transformation, drepresents distance traveled; for the mirror, rotation has no effect;for the wire grid, θ represents the angle of the grid wires with respectto the H-axis; for the rooftop mirror, θ represents the angle of thebetween the roofline and the H-axis; for the birefringent plate, θrepresents the angle between the axis of birefringence and the H-axis,and ξ is half of the phase delay introduced between the orthogonalpolarizations. Description Symbol Jones matrix Stokes Expansion Lineardistance J = (d) $\begin{pmatrix}{\exp\left( {i\; 2\pi\;{d/\lambda}} \right)} & 0 \\0 & {\exp\left( {i\; 2\;\pi\;{d/\lambda}} \right)}\end{pmatrix}\quad$ Ī exp(i2πd/λ) Mirror J _(M) $\begin{pmatrix}1 & 0 \\0 & 1\end{pmatrix}\quad$ Q Wire grid (ref.) J _(WR) $\begin{pmatrix}{\cos^{2}\mspace{14mu}\theta} & {\sin\mspace{14mu}\theta\mspace{14mu}\cos\mspace{14mu}\theta} \\{{- \sin}\mspace{14mu}\theta\mspace{14mu}\cos\mspace{14mu}\theta} & {\sin^{2}\mspace{14mu}\theta}\end{pmatrix}\quad$ ( Q + Ī cos2θ + i V sin2θ)/2 Wire grid (trans.) J_(WT) (θ) $\begin{pmatrix}{\sin^{2}\mspace{14mu}\theta} & {{- \sin}\mspace{14mu}\theta\mspace{14mu}\cos\mspace{14mu}\theta} \\{{- \sin}\mspace{14mu}\theta\mspace{14mu}\cos\mspace{14mu}\theta} & {\cos^{2}\mspace{14mu}\theta}\end{pmatrix}\quad$ (Ī + Q cos2θ − Ū sin2θ)/2 Coord. rotation R (θ)$\begin{pmatrix}{\cos\mspace{14mu}\theta} & {\sin\mspace{14mu}\theta} \\{{- \sin}\mspace{14mu}\theta} & {\cos\mspace{14mu}\theta}\end{pmatrix}\quad$ Ī cosθ + i V sinθ Rooftop Mirror J _(RT) (θ)$\begin{pmatrix}{\cos\mspace{14mu} 2\;\theta} & {\sin\mspace{14mu} 2\;\theta} \\{{- \sin}\mspace{14mu} 2\;\theta} & {\cos\mspace{14mu} 2\;\theta}\end{pmatrix}\quad$ Ī cos2θ + i V sin2θ Biref. plate J _(WP) (θ, ξ)$\begin{pmatrix}{{\cos\mspace{14mu}\xi} - {i\;\sin\mspace{14mu}\xi\mspace{14mu}\cos\mspace{14mu} 2\;\theta}} & {{- i}\;\sin\mspace{14mu}\xi\mspace{14mu}\sin\mspace{14mu} 2\;\theta} \\{{- i}\;\sin\mspace{14mu}\xi\mspace{14mu}\sin\mspace{14mu} 2\;\theta} & {{{- \cos}\mspace{14mu}\xi} - {i\;\sin\mspace{14mu}\xi\mspace{14mu}\cos\mspace{14mu} 2\;\theta}}\end{pmatrix}\quad$

Defining Δ≡4π(d₁−d₂)/λ and setting ξ≡Δ/2, Eq. (5) follows:

$\begin{matrix}{{{\overset{\_}{J}}_{VPM}\left( {{\pi/4},\xi} \right)} = {\frac{\exp\left( {{{\mathbb{i}2\pi}\left( {d_{1} + d_{2}} \right)}/\lambda} \right)}{2}{\begin{pmatrix}{\cos(\xi)} & {- {{\mathbb{i}cos}(\xi)}} \\{{\mathbb{i}}\;\sin} & {- {\cos(\xi)}}\end{pmatrix}.}}} & (5)\end{matrix}$

Eq. (5) forms a basis for derivation of an expression descriptive of avariable-delay polarization modulator 200 placed at an arbitrary angleθ. The definition of θ describes the angle of the grid 215 with respectto H for the input radiation. To form this description, the coordinatesystem is transformed into the coordinate system used for the analysisabove, the transformation for J _(VPM)(π/4) is applied, and then thecoordinate system is transformed back. However, reflections present asubtlety. For an odd number of reflections, the angle θ of the devicefrom the perspective of the outgoing beam is the negative of that fromthe perspective of the incoming beam. This aspect is included as shownbelow. Setting χ=(θ−π/4), the description as stated in Eqs. (6) and (7)is provided:

$\begin{matrix}{{{{\overset{\_}{J}}_{VPM}\left( {\chi,\xi} \right)} = {{{{\overset{\_}{R}}^{\dagger}\left( {- \chi} \right)}{{\overset{\_}{J}}_{VPM}\left( {\pi/4} \right)}{\overset{\_}{R}(\chi)}} = {\frac{\exp\left( {{{\mathbb{i}2\pi}\left( {d_{1} + d_{2}} \right)}/\lambda} \right)}{2}\begin{pmatrix}{{\cos(\xi)} + {{{\mathbb{i}sin}(\xi)}{\cos\left( {2\chi} \right)}}} & {{- {{\mathbb{i}sin}(\xi)}}{\cos\left( {2\chi} \right)}} \\{{{\mathbb{i}sin}(\xi)}{\cos\left( {2\chi} \right)}} & {{- {\cos(\xi)}} + {{{\mathbb{i}sin}(\xi)}{\sin\left( {2\chi} \right)}}}\end{pmatrix}}}},{and}} & (6) \\{{{\overset{\_}{J}}_{VPM}\left( {\theta,\xi} \right)} = {{\exp\left( {{{\mathbb{i}2\pi}\left( {d_{1} + d_{2}} \right)}/\lambda} \right)}{\begin{pmatrix}{{\cos(\xi)} - {{{\mathbb{i}sin}(\xi)}{\cos\left( {2\theta} \right)}}} & {{- {{\mathbb{i}sin}(\xi)}}{\cos\left( {2\theta} \right)}} \\{{{\mathbb{i}sin}(\xi)}{\cos\left( {2\theta} \right)}} & {{- {\cos(\xi)}} - {{{\mathbb{i}sin}(\xi)}{\sin\left( {2\theta} \right)}}}\end{pmatrix}.}}} & (7)\end{matrix}$

In terms of the Stokes parameter basis set, this expression is as shownEq. (8):J _(VPM)(θ,ξ)=exp(i2π(d ₁ +d ₂)/λ)[ Q cos(ξ)−i sin ξ(Ī cos(2θ)+i Fsin(2θ))].  (8)

Aside from a phase factor (irrelevant in measuring power), J _(VPM)= Q J_(WP). Accordingly, the action of the variable-delay polarizationmodulator 200 is equivalent to that of a birefringent plate (having itsbirefringent axis oriented at angle θ, with a delay Δ=2ξ) followed by areflection (represented as the Jones matrix Q).

The matrix in Eq. (8) is unitary and has a determinant of −1. Thus itsMueller representation is expected to describe the symmetries of thePoincaré sphere. By expanding the density matrices in the Pauli matrixbasis both before and after performing similarity transformationscorresponding to the optical system, the Mueller matrix for the systemis determined, as shown below in Eq. (9):

$\begin{matrix}{{{\overset{\_}{M}}_{VPM}\left( {\theta,\Delta} \right)} = {\begin{pmatrix}1 & 0 & 0 & 0 \\0 & {{\cos^{2}2\theta} + {\cos\;{\Delta sin}^{2}2\theta}} & {{- \sin}\; 2{{\theta cos2\theta}\left( {1 - {\cos\;\Delta}} \right)}} & {\sin\; 2{\theta sin\Delta}} \\0 & {\sin\; 2{\theta cos}\; 2{\theta\left( {1 - {\cos\;\Delta}} \right)}} & {{{- \sin^{2}}2\theta} - {\cos\;{\Delta cos}^{2}2\theta}} & {{- \cos}\; 2{\theta sin}\;\Delta} \\0 & {\sin\; 2{\theta sin}\;\Delta} & {\cos\; 2{\theta sin}\;\Delta} & {{- \cos}\;\Delta}\end{pmatrix}.}} & (9)\end{matrix}$The matrix shown in Eq. (9) may be expressed as a product of symmetryoperations on the Poincaré sphere, as given below in Eq. (10):M _(VPM)(θ,Δ)= Γ _(QV) Γ _(QU) R _(V)(2θ) R _(Q)(Δ) R _(V)(−2θ)= Γ _(QV)Γ _(QU) M _(WP)(θ,Δ).  (10)Here, Γ _(QV) represents a reflection about the Q-V plane, Γ _(QU) Γ_(QV) represents a reflection about the Q-U plane, Γ _(V) represents arotation about the V-axis, and Γ _(Q) represents a rotation about theQ-axis. The matrix Γ _(VPM)(θ,Δ) is the Mueller matrix for a wave plate.

§III(b). Polarization Modulation

It is assumed for purposes of this nonrestrictive example that thedetector at the back end of the optical system is sensitive to Stokes Q.This is a statement about the orientation of the analyzer in thepolarimetric system. Strictly speaking, a Q-sensitive detector requiresa differencing of two orthogonal linearly polarized detectors with anorientation that we choose to define as the Q-axis. However, the modeldeveloped below is applicable to the class of polarization detectorstrategies that collect only one linear polarization. Such detectors aretechnically sensitive to Q±I, but to lowest order, or in idealmodulation, I does not couple to the polarization modulation.

The modulator changes the polarization state of the detector asprojected onto the sky. For a single variable-delay polarizationmodulator, the polarization state that the detector measured can becalculated from the second column of the Mueller matrix, as shown belowin Eq. (11):Q _(det) =Q _(sky)(cos²2θ+cos Δ sin²2θ)+U _(sky) cos 2θ sin 2θ(1−cosΔ)+V _(sky)(sin 2θ sin Δ).  (11)

For a variable-delay polarization modulator, θ is fixed and Δ ismodulated. Using a single variable-delay polarization modulator, it isnot possible to completely modulate Q, U and V. An example of this mightbe obtained by setting 0 to π/4. In this case, the expression of Eq.(12) is applicable:Q _(det) =Q _(sky) cos Δ+V_(sky) sin Δ.  (12)

Here, Q and V are modulated, but U is not. This obtains as a result ofthe fact that at this grid 215 angle, U and −U propagate through thesystem separately, without interfering.

An advantage of the variable-delay polarization modulator is that thephase freedom allows a straightforward method for calculating Q and Vacross large frequency bands. A convenient definition set is: Δ≡kδ, suchthat k=2π/λ, where δ represents the total path length difference betweentwo orthogonal polarizations (for traditional Martin-PuplettPupplet beampaths δ=2(d₂−d₁)). The spectrum measured by the polarized detector isthen dependent on δ and is given by Eq. (13) below:Q′(k,δ)=Q(k)cos(kδ)+V(k)sin(kδ).  (13)

For bolometric detectors, the signal is integrated over the passband ofthe instrument, φ(k), as described in Eq. 14:

$\quad\begin{matrix}\begin{matrix}{{Q^{\prime}(\delta)} = {\int_{0}^{\infty}{{Q^{\prime}\left( {k^{\prime},\delta} \right)}{\phi\left( k^{\prime} \right)}{\mathbb{d}k^{\prime}}}}} \\{= {\int_{0}^{\infty}{\left\lbrack {{{Q^{\prime}\left( k^{\prime} \right)}{\cos\left( {k^{\prime}\delta} \right)}} + {{V\left( k^{\prime} \right)}\sin\; k^{\prime}\delta}} \right\rbrack{\phi\left( k^{\prime} \right)}{{\mathbb{d}k^{\prime}}.}}}}\end{matrix} & (14)\end{matrix}$

Taking the Fourier transform of both sides of Eq. (14) yields Eq. (15):

$\begin{matrix}{{{1/{2\left\lbrack {{Q(k)} + {{\mathbb{i}}\;{{iV}(k)}}} \right\rbrack}}{\phi(\lambda)}} = {{1/2}\pi{\int_{\delta_{1}}^{\delta_{2}}{{Q^{\prime}(\delta)}{\mathbb{e}}^{{\mathbb{i}}\; k\;\delta}{{\mathbb{d}\delta}.}}}}} & (15)\end{matrix}$

The real part of the Fourier transform of the interferogram is thespectrum of Stokes Q from the source, while the imaginary part is theStokes V spectrum. Note that broadband modulation relies on sampling alarge enough range of path length differences.

For implementation of a Fourier Transform Spectrometer via aMartin-Puplett interferometer, a horizontal or vertical grid 215 isplaced at the input port of the device 200. The input polarization stateis then purely Q, enabling the internal grid 215 to function as abroadband, frequency-independent beamsplitter. This is equivalent toshorting one of the input ports. Thus, assuming that the polarization ofthe source is small, Q(k)=½(I(k)), and Eq. (15) reduces to theunpolarized spectrum of the source 103. In this case, the device 200does not measure polarization, but relies on the fact that the inputgrid 215 is fixing the polarization to something that is known.

A disadvantage to this architecture is its insensitivity to Stokes U.For experiments using space-borne measurement platforms, U can berecovered by rotation of the spacecraft. For experiments usingterrestrial measurement platforms, sufficient rotation is problematic,and other techniques may be required.

An alternative to measurement platform rotation is to cascade twoinstruments 200 serially. The functional form of the polarization signalmay be found by simply chaining the two Mueller matrices together.

The transfer equation of the system is now S _(sky)= M _(VPM)(θ₁,Δ₁) M_(VPM)(θ₂, Δ₂) S _(det), where modulator₂ is closer to the detector thanmodulator₁. Because the Q_(det) detectors are sensitive only to Q, wesolve for the second column of the resulting matrix, shown below in Eq.(16):

$\begin{matrix}{\left. {Q_{\text{det}} = {{{Q_{sky}\left\lbrack {{\left( {{\cos^{2}2\theta_{1}} + {\cos\;\Delta_{1}\sin^{2}2\theta_{1}}} \right)\left( {{\cos^{2}2\theta_{2}} + {\cos\;\Delta_{2}\sin^{2}2\theta_{2}}} \right)} - {\sin\; 2\theta_{1}\cos\; 2\theta_{1}\sin\; 2\theta_{2}\cos\; 2\theta_{2}}} \right)}\left( {1 - {\cos\;\Delta_{1}}} \right)\left( {1 - {\cos\;\Delta_{2}}} \right)} + {\sin\; 2\theta_{1}\sin\; 2\theta_{2}\sin\;\Delta_{1}\sin\;\Delta_{2}}}} \right\rbrack + {U_{sky}\left\lbrack {{\sin\; 2\theta_{1}\cos\; 2{\theta_{1}\left( {1 - {\cos\;\Delta_{1}}} \right)}\left( {{\cos^{2}2\theta_{2}} + {\cos\;\Delta_{2}\sin^{2}2\theta_{2}}} \right)} - {\left( {{\sin^{2}2\theta_{1}} + {\cos\;\Delta_{1}\cos^{2}2\theta_{1}}} \right)\sin\; 2\theta_{2}\cos\; 2{\theta_{2}\left( {1 - {\cos\;\Delta_{2}}} \right)}} - {\cos\; 2\theta_{1}\sin\; 2\theta_{2}\sin\;\Delta_{1}\sin\;\Delta_{2}}} \right\rbrack} + {{V_{sky}\left\lbrack {{\sin\; 2\theta_{1}\sin\;{\Delta_{1}\left( {{\cos^{2}2\theta_{2}} + {\cos\;\Delta_{2}\sin^{2}2\theta_{2}}} \right)}} + {\cos\; 2\theta_{1}\sin\; 2\theta_{2}\cos\; 2\theta_{2}\sin\;{\Delta_{1}\left( {1 - {\cos\;\Delta_{2}}} \right)}} - {\sin\; 2\theta_{2}\cos\;\Delta_{1}\sin\;\Delta_{2}}} \right\rbrack}.}} & (16)\end{matrix}$

A specific case, given by way of example, where θ₁=π/8 and θ₂=π/4, isnow considered. These angles are chosen to allow full sampling of thePoincaré sphere. Polarized sensitivities for selected pairs of phasedelay settings for the pair of modulators are shown in Table ll. Here itis possible to fully characterize the polarization state, and a simplemethod for doing this is to adopt a single phase delay over the entirebandwidth. In this case, the modulators are set to the desired detectorpolarization sensitivity and a measurement is made. The measurement isrepeated for each state, thus building up information about thepolarization state of the source.

Δ₁ Δ₂ Q_(det) 0 0 Q_(sky) 0 π −Q_(sky) π 0 U_(sky) π π −U_(sky) 0 π/2−V_(sky) π/2 0 ½(Q_(sky) + U_(sky)) + V_(sky)/{square root over (2)} π/2π −½(Q_(sky) + U_(sky)) − V_(sky)/{square root over (2)} π π/2 V_(sky)π/2 π/2 (Q_(sky) − U_(sky))/{square root over (2)}

One of the strengths of this modulator is an ability to modulate quicklybetween different polarization states, in turn providing a benefit byputting the polarization signal above the 1/ƒ knee of the instrumentnoise spectrum. It may also be possible to extend the bandwidth in a waysimilar to the single modulator above. As a result, it is possible toscan these modulators through a range of delays and thus extract thefrequency-independent Stokes parameters.

§III(c). Other Implementations

The architecture exemplified via the illustration of FIG. 2 is not aunique implementation of a variable-delay polarization modulator.Several different arrangements of grids 215 and mirrors correspond toJones matrices that differ only by an absolute phase from those thatdescribe the Martin-Puplett interferometer as derived above. A simpleexample uses a polarizing grid 215 placed in front of a mirror. Thisdesign is similar in structure to a reflecting waveplate, but, in thisdesign, modulation occurs via modulating the grid 215—mirror spacing,rather than by spinning the device. This alternative design for apolarizing interferometer has been previously employed because itprovides compact features and is relatively easy to construct. Thisimplementation is useful in a dual modulator system application becauseit requires significantly less space in the optical path than many othersystems need. However, it does present several drawbacks, including atleast: (i) an inability to achieve a zero path length condition resultsin a slight decrease in effective bandwidth; and (ii) the twopolarizations on the output side are slightly displaced. This effect maybe mitigated slightly by using the modulators at close to normalincidence and by placing the modulators as close as possible to anaperture/pupil.

§III(d). Systematics

In developing a polarization modulator, one must consider thepossibility of instrumental effects introduced by the action of themodulation. In a dielectric half-wave plate, such an effect arises fromthe absorption properties of a birefringent material. Loss tangents forlight polarized along the fast and slow axes are generally different.The result is a modulated signal that appears at twice the rotationalfrequency of the birefringent plate. For the dual variable-delaypolarization modulator system, there are two important effects toconsider, as follows: (i) different settings of the translational stageresult in changes in edge illumination, potentially introducing aspurious polarization signal; and (ii) differential absorption by thegrids 215 and the mirrors 207, 212 of the modulator 200. One of theseconcerns may be obviated by restricting the use of such modulators toslow optical systems in which the beam growth through the modulator isminimal. Another of these concerns is mitigated via use of rooftopmirrors (e.g., rooftop mirrors 207, 212), where the angle of incidenceis the same, for different modulator portions. Thus, the Fresnelcoefficients for each of the two polarizations remains essentiallyconstant during the modulation process.

§III(e). Experimental Results

The description of experimental results comprises four parts. In part(i), the setup is described. In part (ii), experimental procedures areexplained. In part (iii), experimental results are described. In part(iv), resonances are considered.

(i) Setup

FIG. 4 shows a simplified block diagram 400 illustrating avariable-delay polarization modulator 401 useful in the context of thesystem 100 of FIG. 1. The example of the embodiment provided in FIG. 4also is helpful in testing and evaluating performance of thepolarization modulation of a test vehicle exemplifying the principles ofoperation of a Martin-Puplett interferometer in the context of thedisclosed subject matter.

In the block diagram 400 of FIG. 4, the modulator 401 is shown as havingan input port 402 (having a reference plane labeled Γ₁), and an outputport 403 (having a reference plane labeled Γ₂). The input port 402 is302 may be coupled to a feedhorn 404304, which, in turn, may provide asignal 405 (analogous to the signal 102 of FIG. 1) to the modulator 402,where the signal 405 has polarization elements 406 (dashed line) and 407(dot-dashed line) associated therewith.

The beam 405 exiting the horn 404 attached to the input port one 402302subsequently can be redirected via a reflective element 412 and then canpass through a polarizing grid 415 that has its wires oriented at aforty-five degree angle in projection. Each orthogonal polarization 406,407 is 306, 307 can then be launched down a separate arm of the device401301 and reflect off of one of two rooftop mirrors 420, 422, each ofwhich rotates the polarization vector by ninety degrees, as shown abovewith reference to FIG. 2. The beams may recombine at the polarizer 415and may be refocused via a second reflective element 425 into a secondfeed horn 426326 which may be coupled to port two 404.

The experimental setup is shown as being symmetric, and so the reverselight path is identical. The rooftop mirrors 420, 422 are 320, 322 maybe placed on translational stages (not illustrated), such that theirrelative distance (i.e., d₁ and d₂ as noted in FIG. 2) may be adjusted.The frequency-dependent phase that corresponds to this path lengthdifference is the parameter that determines the mapping betweenpolarization states on either end of the device 401. Note that thesystem can support ˜1 mode in each polarization. As a result, Gaussiananalysis was used in designing the optics.

From a microwave circuit perspective, this device 401 may be modeled asa four-port device, with the two ports 402, 404 on either end of thedevice 401 being defined as the vertically and horizontally polarizedelectric field modes. In these experiments, a Hewlett-Packard HP8106Dmillimeter wave vector network analyzer (hereinafter “VNA”) (notillustrated) was used to measure scattering parameters between thesemodes. The calibration reference planes are shown (Γ₁ and Γ₂) in FIG. 4.The VNA may be used to measure the two by two scattering matrices ofpairs of these ports, so in order to reduce contamination of results, arespective orthomode transducer (hereinafter “OMT”) 432, 434 was placedat the back of each respective feedhorn 404, 426 and terminates theunused polarization with a matched load.

For purpose of these measurements, it is useful to regard the end of thefeedhorn 404 attached to one port of the VNA as the source (analogous tothe source 103 of FIG. 1) and that attached to another port of the VNAas the detector. The polarization state of the source for the measureddata (presented infra with respect to FIGS. 5 through 9) was set to bevertically-polarized light (a pure Q state), by orienting the waveguideappropriately. On the detector side (i.e., corresponding to the secondport 403), both V and H states were quantified in successivemeasurements by respectively omitting and adding a ninety-degree twistto the WR-10 waveguide between the OMT 434 and the Γ₂ calibration point403. The loss of the twist 436 is measured as 0.15 dB. The calibrateddifference between the power associated with H and V gives a measurementof Stokes Q at the detector.

The bandwidth of the test setup is approximately 78-115 GHz. At the lowend of the band, the band edge is defined by that of the W-band feedhorns 404, 426, and at the high end, it is defined by return loss due tothe OMTs 432, 434.

The experimental data presented via FIGS. 5 through 9, as described in§III(f), infra, employ a pair of W-band feed horns 404, 426 (25-27 dBi)that are collimated by ellipsoidal mirrors 412, 425 (ƒ=25 cm). Each ofthe two rooftop mirrors 420, 422 reflect 320, 322 reflects a componentof polarizations. The mirrors are mounted on transports employed toadjust the path lengths of the individual polarizations. The polarizinggrid is mounted such that the wires make an angle of forty-five degreeswith the roof lines of (i.e., lines between corresponding or opposedpoints on the surfaces of) the rooftop mirrors 412, 425 in projection.The dashed and dotted lines show positions of the beam radius (8.7 dBedge taper and 20 dB edge taper, respectively) of a Gaussian beampropagating through the structure for a 26 dBi feed and a wavelength ofthree millimeters (100 GHz). A dotted outline illustrates location for aninety degree twist 436 that converts the sensitivity of port two from Vto H.

(ii). Experimental Procedure

The zero path length position is found by first measuring the signal inthe V direction at an experimental condition where the S₂₁ parameter wasflat across the band. A first null condition was then used to achievefine adjustment. V and H were measured for twenty-seven combinations ofpositions of the two mirrors (i.e., such as the rooftop mirrors 207, 212of FIG. 2 or 420, 422 of FIG. 4) having path differences (e.g., such asd₁−d₂) corresponding to twenty-four degree steps in phase for λ=3 mm.FIGS. 5 through 8 show four measured spectra. These plots include theexpected transmission spectra (Hα(1−cos Δ) and Vα(1+cos Δ)), adopting aglobal gain of 0.9 dB to account for the expected loss beyond thecalibration port. The return loss of the system is about twenty-six dBand can been seen in the H component in FIG. 4. The transmissionefficiency of the horns 404, 426 used here is not constant across theband, and tends to roll off at low frequencies.

(iii). Experimental Results

The experimental setup is described mathematically by Eq. (12). In thiscase,

$\quad\begin{matrix}\begin{matrix}{Q_{\text{det}} = \left( {{H(\Delta)} - {{{fV}(\Delta)}/\left( {{H(\Delta)} + {{fV}(\Delta)}} \right)}} \right.} \\{{= {{Q_{source}\cos\;\Delta} + {V_{source}\sin\;\Delta}}},}\end{matrix} & (17)\end{matrix}$where Δ=4π(d₂−d₁)/λ. Here, H(A) and V(A) are the powers corresponding toS₂₁ when the twist is included and excluded, respectively. For eachfrequency, the relative gain factor, ƒ, is calculated by fitting for theaverage values of the signals in the H and V configurations and thentaking the ratio.

For every frequency, it is possible to measure Stokes Q and V. Theresult of this analysis is shown in FIG. 7. We find that that averageStokes parameters measured over the 78-110 GHz band are Q=−1.002±0.003and V=0.001±0.013. There is some non-zero power in Stokes V near thehigh end of the band. It is unclear whether this is due to a systematiceffect, or to an unknown source polarization.

(iv). Resonances

In this setup, proper termination of the unused port at both theentrance (202,402) and exit (204, 403) apertures is essential, as evensmall reflections can introduce resonances. These resonances are anindication of level of uncertainty of phase control of the radiationpropagating through the device 200, 300, 400. In turn, such uncertaintyleads directly to a frequency-dependent random mixing between the Q andV polarization states, and hence to a decrease in the precision of theMartin-Puplett interferometer as a polarimeter.

To address such factors, it is noteworthy that systematic “noise” may bemitigated via various levels of termination of the unused polarization.For example, addition of the OMTs 432, 434 in the signal chain reducednoise in the transmission from three dB to one dB. This setup alsoemployed a conventional horizontal grid (not illustrated), analogous togrid 215, 315, 415, at the mouth of the appropriate feedhorn 404, 426 toredirect any residual H component to a conventional eccosorb beam dump(also not illustrated). Such reduced the noise in transmission toone-fourth dB and further reduced the average coupling Q into V fromfour percent to under one percent.

On a telescope, this problem is mollified by the fact that the source103 component is nearly perfectly terminated in the sky. This greatlyreduces phase uncertainties in the system, as well as need for excessivepolarization filtering.

An additional consideration is that the ability of these variable-delaypolarization modulator devices 200, 300, 400 to operate at roomtemperature also favors application as calibration devices. An inputpolarized signal is easily transformed to test polarization response ofa precision polarization sensor. These variable-delay polarizationmodulator devices 200, 300, 400 are capable of transforming an initiallylinearly polarized state into an elliptical polarization state.

§III(f). Experimental Results vs. Simulations

FIGS. 5 through 9 summarize experimental results obtained as describedabove with respect to §§(i) through (iv). The thick solid line in eachof these graphical depictions represents the spectrum of the vertical(V) linear polarization as measured at port two of the vector networkanalyzer. The thin solid line accompanying this represents the spectrumof the corresponding horizontal (H) linear polarization, also measuredat port two of the vector network analyzer. The H polarization componentis measured by adding a ninety-degree twist in the WR-10 waveguideattached to port two of the VNA. Predictions for H and V provided viathe theory and models discussed above are plotted as thick and thindashed lines, respectively. FIGS. 5 through 8 employ axes calibrated indecibels (dB, ordinate), and frequency, expressed in gigaHertz(abscissa).

FIG. 5 provides a graph 500 of signal transmission amplitudes incomparison to respective predicted expectation values 502, 504 forhorizontal (H) 512 and vertical (V) 514 polarization components of asignal (e.g., signal 102 of FIG. 1). The relative separation d₁-d₂ ofthe rooftop mirrors 420, 422 (FIG. 4) for the data displayed in FIG. 5is a negative thirteen micrometers.

FIG. 6 provides a graph 600 of signal transmission amplitudes incomparison to estimated values 602, 604 for respective horizontal (H)612 and vertical (V) 614 polarization components of the signal of FIG.6. The relative separation d₁-d₂ of the rooftop mirrors 420, 422 (FIG.4) for the data displayed in FIG. 6 is five hundred thirteenmicrometers.

FIG. 7 provides a graph 700 of signal transmission amplitudes incomparison to estimated values 702, 704 for respective horizontal (H)712 and vertical (V) 714 polarization components of the signal of FIG.7. The relative separation d₁-d₂ of the rooftop mirrors 420, 422 (FIG.4) for the data displayed in FIG. 7 is a negative one thousand thirteenmicrometers.

FIG. 8 provides a graph 800 of normalized Stokes parameters Q 802, 804for respective horizontal (H) 812 and vertical (V) 814 polarizationcomponents of the signal of FIG. 8. The relative separation d₁-d₂ of therooftop mirrors 420, 422 (FIG. 4) for the data displayed in FIG. 8 is anegative one thousand thirteen micrometers.

FIG. 9 provides a graph 900 of normalized Stokes parameters Q 902 and V904 as a function of frequency, by fitting to the twenty-seven positionsfor the rooftop mirrors 420, 422 of FIG. 4. Mean values for Q 902 and V904 across the seventy-eight to one hundred fifteen GHz band are−1.002±0.003 and 0.001±0.013, respectively.

§III(g). Polarization Matrix Methods

The overview of polarization matrix methods is developed in three parts.In part (i), Jones matrices are described. In part (ii), densitymatrices are described. In part (iii), Mueller matrices are described.

(i). Jones Matrices

Jones matrices provide convenient modeling tools for analysis ofradiation as it propagates through an optical system in which knowledgeof phase is relevant. For an ideal case, it is assumed that all portsare impedance matched, and so no cavities are formed. This formulationis applicable for coherent radiation; however, it may be extended usingthe closely-related formalism of density matrices to treat situationsinvolving partially-polarized radiation or light.

Jones matrices are two by two matrices expressing information regardinghow orthogonal electrical field components transform in an opticalsystem. The input Jones vector is expressed as shown below in Eq. (18):

$\quad\begin{matrix}\begin{matrix}{\left. E \right\rangle = \left( \frac{E_{x}}{E_{y}} \right)} \\{\equiv {\left( \frac{E_{H}}{E_{V}} \right).}}\end{matrix} & (18)\end{matrix}$The output vector from an optical system may then be represented by|E_(f)>= J|E_(i)<, where J is the vector transformation introduced bythe optical system. The power measured at a detector at the back end ofsuch a system may be expressed as shown in Eq. (19), below:

E _(f) | J _(det) |E _(f)

=

E _(i) | J ^(†) J|E _(i)

.  (19)The matrix J _(det) depends on properties of the detector used to makethe measurement. In Jones matrix representation, Stokes parameters arerepresented by appropriate Pauli matrices and the identity matrix, asshown below:

$\quad\begin{matrix}{{{{\overset{\_}{I} \equiv \overset{\_}{\sigma_{0}}} = \begin{pmatrix}1 & 0 \\0 & 1\end{pmatrix}};}{{{\overset{\_}{Q} \equiv \overset{\_}{\sigma_{1}}} = \begin{pmatrix}1 & 0 \\0 & {- 1}\end{pmatrix}};}{{{\overset{\_}{U} \equiv \overset{\_}{\sigma_{2}}} = \begin{pmatrix}0 & 1 \\1 & 0\end{pmatrix}};}{{\overset{\_}{V} \equiv \overset{\_}{\sigma_{3}}} = {\begin{pmatrix}0 & {- {\mathbb{i}}} \\{- {\mathbb{i}}} & 0\end{pmatrix}.}}} & (20)\end{matrix}$In these representations, the bar over the Stokes symbol indicates theJones matrix representation. An un-barred Stokes symbol representsmeasurable power (e.g., Q=

E_(f)| J _(det)|E_(f)

). The measured power in each of the Stokes parameters is given via Eqs.(21) through (24) below:

$\quad\begin{matrix}\begin{matrix}{I = \left\langle {E{\overset{\_}{I}}E} \right\rangle} \\{{= {E_{H}^{2} + E_{V}^{2}}};}\end{matrix} & (21) \\\begin{matrix}{Q = \left\langle {E{\overset{\_}{Q}}E} \right\rangle} \\{{= {E_{H}^{2} - E_{V}^{2}}};}\end{matrix} & (22) \\\begin{matrix}{U = \left\langle {E{\overset{\_}{U}}E} \right\rangle} \\{{= {2{\Re\left( {E_{H}^{*}E_{V}} \right)}}};}\end{matrix} & (23) \\\begin{matrix}{V = \left\langle {E{\overset{\_}{V}}E} \right\rangle} \\{= {2{{{??}\left( {E_{H}^{*}E_{V}} \right)}.}}}\end{matrix} & (24)\end{matrix}$

Eqs. (21) through (24) connect the Jones matrix formulation of theStokes parameters to their familiar definitions. These four Stokesmatrices have the following multiplicative properties: if ( σ₀ , σ₁ , σ₂, σ₃ )≡(Ī, Q, Ū, V), σ₀ σ_(α) = σ_(α) σ₀ = σ_(α) for αε(0, 1, 2, 3) andσ_(i) σ_(j) =Σ_(i)ε_(jkl)i σ_(l) +δ_(jk) σ₀ and for j, k, lε(1, 2, 3).These four matrices form a convenient basis for expressing Jonesmatrices.

(ii). Density Matrices

Density matrices are two by two matrices which fully characterize thepolarization state of the light. For the general case of partiallypolarized light, polarization arises because of time-averaged(statistical) correlations between the electric field components. Adensity matrix D is represented as shown below with reference to Eq.(25):

$\begin{matrix}{\overset{\_}{D} = {\begin{pmatrix}\left\langle {E_{x}^{*}E_{x}} \right\rangle & \left\langle {E_{x}^{*}E_{y}} \right\rangle \\\left\langle {E_{y}^{*}E_{z}} \right\rangle & \left\langle {E_{y}^{*}E_{y}} \right\rangle\end{pmatrix}.}} & (25)\end{matrix}$

The brackets < > indicate a time average. When the density matrix isexpressed as a linear combination of Pauli matrices, D=I σ₀ +Q σ₁ +U σ₂+V σ₃ , the coefficients are the Stokes parameters.

Transformation of the polarization state by an optical system may bedescribed via a similarity transformation, D′= J′ D J. Here, J is theJones matrix describing the optical system. For the purposes of thisdisclosure, the manner in which the polarization state of the detectorsmaps onto the sky is relevant. Accordingly, D _(sky)= J′ D _(det) J. Thetransformation of the density matrix D and the expression for totalpower in the Jones formalism as expressed in Eq. (19) are notablysimilar.

(iii). Mueller Matrices

Mueller matrices are four by four matrices. In an analogy to specialrelativity, the inhomogeneous Lorentz group may be represented by agroup of four by four real matrices acting on a Stoke vector, S=(I, Q,U, V), and these matrices are known as Mueller matrices.

In the prior discussion, no limitations had been placed on J, the matrixdescribing the optical system being modeled. When the magnitude of thedeterminant of J is unity, there is a homomorphism between the group oftwo by two matrices having |det J|=1 and the Poincaré or inhomogeneousLorentz group. With respect to the subject matter of this disclosure,the Mueller matrix that maps the Stokes parameters at the detector tothe sky are of note, viz., S _(sky)= M S _(det).

For polarization modulation, the instances for which the Jones matricesdescribing the optical system are unitary are of interest. For thiscase, Stokes I decouples from the other Stokes parameters, and thequantity P²=Q²+U²+V² is preserved. This subgroup may be represented bythree by three orthogonal submatrices representing symmetries on thesurface of a sphere in space having Stoke I, Q and V as axes. Thissphere is the Poincaré sphere.

When the group of density matrices are restricted to those with positivedeterminants, the system is described by SU(2), and so there ishomomorphism between this group and the SO(3) group of rotations on thePoincaré sphere. These are the groups that are important to a waveplate; however, the physical reflection involving in the VPMarchitecture introduces a negative determinant, resulting incombinations of rotations on the Poincaré sphere.

§III(h). Summary

Techniques for polarization modulation in which n phase delays betweenlinear orthogonal polarizations are placed in series with arbitraryrelative orientations as are described and modeled in this disclosure.The n=1 and n=2 cases are specifically considered, and it is noted thatfor appropriate relative orientations, it is possible to fully modulatethe polarization in the n=2 case. In the far-infrared through millimeterwhere bandpasses are typically Δλ/λ˜0.1, this device can be used in asimilar manner to a half-wave plate. Broader passbands (Δλ/λ˜0.3) may beaccommodated using more complex modulation schemes. This architectureenables construction of a modulator that may be made robust, broadband,and easily tunable to different wavelengths to provide frequencydiversity. In addition, it permits complete determination of thepolarization state of the incoming radiation by measurement of Stokes Q,U and V.

§V. CONCLUSION

A polarization sensitive receiver system is described. Although specificembodiments have been illustrated and described herein, it will beappreciated by those of ordinary skill in the art that any arrangementwhich is calculated to achieve the same purpose may be substituted forthe specific embodiments shown. This disclosure is intended to cover anyadaptations or variations. For example, although described in proceduralterms, one of ordinary skill in the art will appreciate thatimplementations can be made in a procedural design environment or anyother design environment that provides the required relationships.

In particular, one of skill in the art will readily appreciate that thenames or labels of the elements and apparatus are not intended to limitembodiments. Furthermore, additional processes and/or apparatus can beadded to the components, functions can be rearranged among thecomponents, and new components to correspond to future enhancements andphysical devices used in embodiments can be introduced without departingfrom the scope of embodiments.

1. A signal conditioning module configured for insertion into aphotometric system to enable the photometric system to operate as apolarimeter, the signal conditioning module comprising: a plurality ofcascaded variable delay polarization modulators, each of the pluralityof modulators comprising: an input port; a first arm comprising a firstreflector and first rooftop mirror arranged in opposed relationship, thefirst reflector directing an input radiation signal from the input porttowards the first rooftop mirror; an output port; a second armcomprising a second reflector and second rooftop mirror arranged inopposed relationship, the second reflector guiding a signal from thesecond rooftop mirror towards the output port to provide an outputradiation signal; a grid placed between the first reflector and thefirst rooftop mirror, and also placed between the second reflector andthe second rooftop mirror; and translation apparatus for adjustment ofrelative optical path length vis-à-vis the first arm, the second arm andthe grid.
 2. The module of claim 1, wherein the module is configured toprovide frequency diversity.
 3. The module of claim 1, wherein themodule comprises two variable delay polarization modulators.
 4. Themodule of claim 1, wherein the first and second reflectors compriseellipsoidal mirrors.
 5. The module of claim 1, wherein the firstreflector provides a collimated signal to the first arm.
 6. The moduleof claim 1, wherein the first reflector collimates the input radiationsignal to provide a collimated signal to the first arm, and wherein thesecond reflector focuses the output radiation signal from the second arminto an output waveguide.